Hensel minimality: Geometric criteria for ℓ-h-minimality

Recently, Cluckers, Halupczok and Rideau-Kikuchi developed a new axiomatic framework for tame non-Archimedean geometry, called Hensel minimality. It was extended to mixed characteristic together with the author. Hensel minimality aims to mimic o-minimality in both strong consequences and wide applicability. In this article, we continue the study of Hensel minimality, in particular focusing on ω-h-minimality and ℓ-h-minimality, for ℓ a positive integer. Our main results include an analytic criterion for ℓ-h-minimality, preservation of ℓ-h-minimality under coarsening of the valuation and ℓ-dimensional dimensional geometry.